 Trivial coefficients detect the cohomological dimension of a pro-p group

For a pro-p group $P$, vanishing of $H^{n+1}(P,{\mathbb{F}}_p)$ implies vanishing in that degree for every discrete $p$-primary module. Finite such modules have composition factors equal to the trivial module ${\mathbb{F}}_p$, so the long exact sequence gives the result by induction. General modules are unions of finite stable submodules, and continuous cohomology commutes with filtered unions. Dimension shifting gives ${\mathrm{cd}}_p(P)\le n$. The converse is immediate from the definition.